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Isometry group of Euclidean space
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n). The Euclidean group E(n) comprises all translations, rotations, and reflections of ; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space. A Euclidean isometry can be direct or indirect, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(n) and E+(n), whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections. These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.

The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case 1=n = 2, and 6 for 1=n = 3. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.

The direct isometries (i. e., isometries preserving the handedness of chiral subsets) comprise a subgroup of E(n), called the special Euclidean group and usually denoted by E+(n) or SE(n). They include the translations and rotations, and combinations thereof; including the identity transformation, but excluding any reflections. The isometries that reverse handedness are called indirect, or opposite. For any fixed indirect isometry R, such as a reflection about some hyperplane, every other indirect isometry can be obtained by the composition of R with some direct isometry. Therefore, the indirect isometries are a coset of E+(n), which can be denoted by E−(n). It follows that the subgroup E+(n) is of index 2 in E(n).

The natural topology of Euclidean space implies a topology for the Euclidean group E(n). Namely, a sequence fi of isometries of is defined to converge if and only if, for any point p of , the sequence of points pi converges. From this definition it follows that a function is continuous if and only if, for any point p of , the function defined by fp(t) = (f(t))(p) is continuous. Such a function is called a "continuous trajectory" in E(n). It turns out that the special Euclidean group SE(n) = E+(n) is connected in this topology. That is, given any two direct isometries A and B of , there is a continuous trajectory f in E+(n) such that f(0) = A and f(1) = B. The same is true for the indirect isometries E−(n). On the other hand, the group E(n) as a whole is not connected: there is no continuous trajectory that starts in E+(n) and ends in E−(n). The continuous trajectories in E(3) play an important role in classical mechanics, because they describe the physically possible movements of a rigid body in three dimensional space over time. One takes f(0) to be the identity transformation I of , which describes the initial position of the body. The position and orientation of the body at any later time t will be described by the transformation f(t). Since f(0) = I is in E+(3), the same must be true of f(t) for any later time. For that reason, the direct Euclidean isometries are also called "rigid motions".

The Euclidean groups are not only topological groups, they are Lie groups, so that calculus notions can be adapted immediately to this setting.